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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, 683, 5164]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, 683, 5164]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, 683, 5164]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 46255i have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(5\)\(1 + T\)
\(11\)\(1 - T\)
\(29\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - 2 T + 2 T^{2}\) 1.2.ac
\(3\) \( 1 + T + 3 T^{2}\) 1.3.b
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(13\) \( 1 - 5 T + 13 T^{2}\) 1.13.af
\(17\) \( 1 + 4 T + 17 T^{2}\) 1.17.e
\(19\) \( 1 + T + 19 T^{2}\) 1.19.b
\(23\) \( 1 + 2 T + 23 T^{2}\) 1.23.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 46255i do not have complex multiplication.

Modular form 46255.2.a.i

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{2} - q^{4} + q^{5} + 3 q^{8} - 3 q^{9} - q^{10} + q^{11} + 2 q^{13} - q^{16} - 6 q^{17} + 3 q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 46255i

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
46255.e4 46255i1 \([1, -1, 1, 683, 5164]\) \(59319/55\) \(-32715282655\) \([2]\) \(24192\) \(0.70483\) \(\Gamma_0(N)\)-optimal
46255.e3 46255i2 \([1, -1, 1, -3522, 48896]\) \(8120601/3025\) \(1799340546025\) \([2, 2]\) \(48384\) \(1.0514\)  
46255.e2 46255i3 \([1, -1, 1, -24547, -1439674]\) \(2749884201/73205\) \(43544041213805\) \([2]\) \(96768\) \(1.3980\)  
46255.e1 46255i4 \([1, -1, 1, -49777, 4285854]\) \(22930509321/6875\) \(4089410331875\) \([2]\) \(96768\) \(1.3980\)